Differential equations and differential calculus in Montel spaces

Abstract

A topological vector space is a set which is equipped with a geometric structure and a topological structure. It is natural to ask if this is sufficient to derive an analytic structure, or if one has to impose it from without as in the case of manifolds. By an analytic structure, we mean the operations of differentiation and integration, and their corresponding properties. It is well established that integration can be defined reasonably (for a summary of the vast amount of literature on this subject, see Hyers [7]). In the case of differentiation, until recently, little has been done beyond the case of Banach spaces. In the latter case, the program has been completed by Hildebrandt and Graves [6], who developed the calculus, Kerner [8], who characterized the perfect differentials, Michal and Elconin [9], who extended the Picard existence and uniqueness theorems for differential equations, and Dieudonne [4], who showed that the weaker Peano existence theorem, which assumes only continuity, is false for Banach spaces. To begin the extension to non-normed spaces, Gil de Lamadrid [5], gave a definition of the derivative and started the development of the calculus. In the present paper, for the case of Montel spaces (to be defined below), we extend the calculus, characterize the perfect differentials and prove existence and uniqueness theorems for differential equations. In this paper, R will refer to the set of real numbers, 0 the empty set, and the symbol, 1, will indicate the completion of a proof. Iff and g are two functions with the property that the range of g is contained in the domain of f, then f o g will indicate the composition of these two functions. If A, B, are two sets, then will mean the set of all functions (in whatever function space is being considered) which map A into B. If C is another set, then o will be the set of all compositions,f og, withf ( and g E . It is obvious that o c .